An Estimate of Canonical Dimension of Groups Based on Schubert Calculus

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K L aho eto 2a]) Section of raph 0 . K ngnrl it general, In ? K and E defined L but , L rt )? . the definition of the essential 0-dimension of a scheme, which is known to coincide with the (standard in algebraic geometry) dimension, see [13, Proposition 1.2]. This can be viewed as a motivation for the word “dimension”. (But essential dimension is not only defined for schemes, and in broader generality it becomes a much more nontrivial notion.) Another motivation for canonical (0-)dimension comes from incompressible varieties, but this motivation is only valid for the canonical (0-)dimension of smooth complete schemes. The definition that we are going to give next, the canonical 0-dimension of an algebraic group, and that will be used in the main theorem of this text, does not involve the canonical (0-)dimension of smooth complete schemes, so this motivation will be useless for us. One can find details for this motivation in [8, Section 2]. The second object we need to define before we can define the canonical 0-dimension of a group is a torsor of a group. All algebraic groups in this text are affine. All reductive, semisimple, and simple groups in this text are smooth. Torsors of algebraic groups (over a point) are, informally speaking, homogeneous spaces that are “as large as the group itself”. This notion is mostly interesting over non algebraically closed fields. Definition 1.2 ([14, Section 3a]). Given an algebraic group G, a G-torsor over a point (or simply a G-torsor) is a scheme E with an action ϕ: G × E → E such that (ϕ, pr2): G × E → E × E, where pr2 is the projection to the second factor, is an isomorphism. It is known that all torsors of affine algebraic groups over a point are affine. Finally, the canonical 0-dimension of an algebraic group understood as a group measures how hard it is to get rational points in torsors, informally speaking, related to the group. Precisely: Definition 1.3 ([14, Section 4g]). given an algebraic group G over a field F , the canonical 0-dimension of G understood as a group (notation: cd0(G)) is

cd0(G)= max max cd0(E). K=a field containing F E=a GK -torsor The definition of canonical dimension of an algebraic group understood as a group in [9, Introduction] repeats this definition almost exactly, with the only difference being that instead of cd0(E) it uses the definition of canonical dimension of E understood as a scheme from the paper [9] itself. But as we already mentioned above, it is known that these two notions are known to be equivalent for torsors of split reductive groups. So, Definition 1.3 is also equivalent to the definition of canonical dimension of a group from [9, Introduction] for split reductive groups. All groups whose canonical dimension we are going to estimate in this text are split reductive (and even simply connected semisimple), so these results also estimate the canonical dimension in the sense of [9, Introduction]. To formulate the main goal of this text precisely, we need to introduce some more notation and terminology. Given a split semisimple algebraic group G and a Borel subgroup B, the corresponding Weyl group W , and the element w0 ∈ W of maximal length, for each w ∈ W we denote the Schubert −1 variety Bw0w B/B ⊆ G/B by Zw. This Zw is a Schubert divisor if and only if w is a simple reflection, and we denote all Schubert divisors by D1,...,Dr. It is known that the classes [Zw] ∈ CH(G/B) for all w ∈ W form a free set of generators of n1 nr CH(G/B) as of an abelian group. We say that a product of classes of Schubert divisors [D1] ... [Dr] is multiplicity-free if there exists w ∈ W such that the coefficient at [Zw] in the decomposition of n1 nr [D1] ... [Dr] into a linear combination of Schubert classes equals 1. Now we can formulate the goal of this text precisely. Our goal is to sketch the proof of the following theorem. Theorem 1.4. Let G be a split semisimple simply connected algebraic group over an arbitrary field, let B be a Borel subgroup, let r be the rank of G, and let D1,...,Dr ⊂ G/B be the Schubert divisors n1 nr corresponding to the r simple roots of G. If [D1] ... [Dr] is a multiplicity-free product of Schubert divisors, then cd0(G) ≤ dim(G/B) − n1 − ... − nr. As a corollary of this theorem and [5, Theorem 11.5], we will immediately get the following:

Corollary 1.5. Let G be a split semisimple simply connected algebraic group of type Er. Then cd0(G) ≤ 17, 37, or 86 for r =6, 7, or 8, respectively.

2 The canonical dimension of simply connected split groups of type Ar and Cr is known to be zero. For types Br and Dr, the canonical dimension was estimated (and computed exactly if r is a power of 2) by N. Karpenko in [10]. The paper [10] also relates cohomology of flag varieties of orthogonal groups (more precisely, orthogonal Grassmannians, not full flag varieties) to canonical dimension, and in this text, we are going to follow the ideas of several proofs from [10]. For type G2, the canonical dimension (of a split simply connected group) is known and equals 3, see [1, Example 10.7]. For type F4, no nontrivial upper bounds on the canonical dimension are known. In types Er, the most difficult part of obtaining Corollary 1.5 was to understand which products of Schubert divisors are multiplicity-free (and this was understood in [5] by the author). The part of the argument establishing relation between Schubert calculus and canonical dimension (in other words, the proof of Theorem 1.4 itself) was known to the experts (or at least they believed that the argument is doable this way). However, we were unable to find an exposition suitable for more general mathematical audience. The present paper contains such an exposition.

Acknowledgments I thank Kirill Zaynoulline for bringing my attention to the problem. I thank Nikita Karpenko for useful discussions and explanations about theory of torsors and canonical dimension and about his paper [10]. I also thank Vladimir Chernousov and Alexander Merkurjev for useful discussions about torsors and theory of Galois descent. I also thank the following people for discussions about intersection theory and algebraic geometry in general: Stephan Gille, Marat Rovinskiy, Nikita Semenov, Alexander Vishik, and Bogdan Zavyalov.

Funding This research was partly supported by the Paciﬁc Institute the Mathematical Sciences fellowship. The author also thanks Max Planck Institute for Mathematics in Bonn for its ﬁnancial support and hospitality.

2 Isomorphism of Picard groups under scalar expansion

We always denote by idX : X → X, where X is a scheme, the identity map. To start proving Theorem 1.4, we first need to define the quotient of a torsor modulo a Borel subgroup. The definition we are going to use is not very intuitive, but it is used in papers on canonical dimension (for example, in [9]). Definition 2.1. Let G be a semisimple split algebraic group over a field K, let B be a Borel subgroup, and let E be a G-torsor. The quotient of the torsor modulo the Borel subgroup (notation: E/B) is the categorical quotient (see [16, Definition 0.5]; “categorical” is in the category of all separated schemes of finite type over K) of E × G/B modulo the diagonal action of G. In fact, it can be proved that such a quotient is also a categorical quotient of E modulo B, but we will not use this. The existence of such a categorical quotient (E × G/B)/G is known, is stated in [6, Proposition 12.2], and can be proved using Galois descent theory. It is known that such a quotient E/B is smooth, absolutely irreducible, and projective. Given this definition, we can say that the first and the most technically difficult step in proving Theorem 1.4 is to prove the following proposition. Proposition 2.2. Let G be a semisimple split algebraic group over a field K, let B be a Borel subgroup, and let E be a G-torsor. Let K1 be an extension of K. Then the map of Picard groups induced by field extension Pic(E/B) → Pic((E/B)K1 ) is an isomorphism. The proof of this proposition makes a lot of use of Galois descent theory. Let us recall the basic notions and facts of this theory (or of the form of this theory that we need). We will need three categories. The first category, SchK is the category of (separated and of finite type, as everywhere in the text) schemes over a field K.

3 To define the second category, suppose we have two fields, K ⊆ L. First, we need to recall the definition of the functor of restriction of scalars from SchL to SchK (notation: −|K ). If X is an object of SchL, we say that X with scalars restricted from L to K is the scheme that has the same topological space as X, the same ring of regular functions on each open subset as an abstract ring, but for the algebra structure, we view this ring as a K-algebra rather than an L-algebra (the multiplication by elements of

K is given by the embedding K ⊆ L). We denote this scheme by X|K. And if f ∈ MorSchL (X, Y ), then one can check directly that the same map of topological spaces as in f, together with the same map of abstract rings for each open subset of Y (= each open subset of Y |K ) as in f, satisﬁes the deﬁnition of a morphism of K-schemes from X|K to Y |K . Now we can say that the second category, which we will call the category of K,L-schemes (notation:

SchK,L), has schemes over L as objects, and the set MorSchK,L (X, Y ), where X and Y are L-schemes, is the set of morphisms of K-schemes from X|K to Y |K . The third category will be introduced a bit later. Example 2.3. Let K ⊆ L be a finite Galois extension of fields, and let σ ∈ Gal(L/K). Let X = Spec L. −1 Then K[X|K] = L, and σ : L → L is an automorphism of this K-algebra. So, we have a dual automorphism of the K-scheme X|K , which we denote by σ∗ ∈ MorSchK,L (Spec L, Spec L). We keep the notation σ∗ until the end of the text. Definition 2.4. Let K ⊆ L be a finite Galois extension of fields with Galois group Γ, and let σ ∈ Γ. A morphism f : X → Y in SchK,L is called σ-semilinear if the following diagram (in SchK,L) is commutative: f X / Y

σ∗ Spec L / Spec L The vertical arrows are the restrictions of scalars of the structure morphisms. Clearly, under the conditions of this definition, if f (resp. g) is a σ (resp. τ)-semilinear morphism, then g ◦ f is a τσ-semilinear morphism. It is also clear that 1-semilinear morphisms are exactly the restrictions of scalars of the morphisms in SchL. Definition 2.5. Let K ⊆ L be a finite Galois extension of fields with Galois group Γ. We will say that we have a Galois-semiaction of Γ on an L-scheme X (or that Γ Galois-semiacts on X) if we have an action ψ : Γ × X|K → X|K (here Γ is understood as an algebraic group over K) such that for each

σ ∈ Γ, the automorphism ψσ = ψ|{σ}×X|K of X|K, understood as an automorphism of X in SchK,L, is σ-semilinear. We say that a finite affine open covering of X is Γ-stable if Γ preserves (normalizes) each of the open sets. Now we are ready to define the third category we need to formulate basic facts of Galois descent theory. Given a Galois extension of fields K ⊆ L with Galois group Γ, we define the category of stable L-schemes with semiaction of Γ (notation: (StSchL, Γ)). Its objects are pairs (X, ψ), where X is an L-scheme, and ψ : Γ × X|K → X|K is a Galois-semiaction such that X admits a Γ-stable finite affine open covering. The morphisms are morphisms in SchL that become Γ-equivariant after the restriction of scalars to K. Now recall that if a finite group acts on a scheme (now this is going to be a scheme over the smaller field, K), and there is a stable finite affine open covering for this action, then the categorical quotient always exists, and can be constructed, for example, as the orbit space of the action. So, for a Galois extension K ⊆ L with group Γ, we can define the Galois descent functor DecK : (StSchL, Γ) → SchK as follows: an object (X, ψ) is mapped to the categorical quotient X/Γ, and the morphisms are mapped using the universal property of the categorical quotient. We can also define the Galois upgrade functor ·L,Γ : SchK → (SchL, Γ). On the objects, it maps a K-scheme Y to (YL, ϕ), where the semiaction ϕ is defined on the affine charts as follows: if U is an open affine chart of Y , σ ∈ Γ, then ϕ(Γ, (UL)|K ) = (UL)|K (recall that the restriction of scalars does not change the topological space).

4 ∗ −1 And if f ⊗ λ ∈ L[UL]= K[U] ⊗ L, then (ϕ|{σ}×(UL)|K ) (f ⊗ λ) = f ⊗ σ (λ). On the morphisms, the Galois upgrade functor is just expansion of scalars. Given these two functors, let us state the main theorem of Galois descent theory Theorem 2.6. Let K ⊆ L be a Galois extension with Galois group Γ. The Galois descent and upgrade functors are mutually quasiinverse equivalences of categories SchK ↔ (StSchL, Γ). Proof. Well-known. Uses the explicit construction of the categorical quotient modulo Γ as an orbit space. To apply this theory to algebraic groups, we first need to understand how direct products work in SchK,L and in (StSchL, Γ). The direct products in SchL and in SchK,L are different. However, the following lemma shows that direct products from SchL are useful in SchK,L if we work with semilinear morphisms. Lemma 2.7. Let K ⊆ L be a Galois extension of fields with Galois group Γ. Let X and Y be L-schemes, let Z be their product in SchL, and let p1 ∈ MorSchL (Z,X) and p2 ∈ MorSchL (Z, Y ) be the standard projections. Then for every L-scheme T , for every σ ∈ Γ, and for every two σ-semilinear morphisms f : T → X and g : T → Y there exists a unique σ-semilinear morphism h: T → Z such that p1|K ◦ h = f and p2|K ◦ h = g. Proof. Well-known.

Suppose, for K, L, Γ, X, Y , Z, p1, and p2 as in the lemma, we have two σ-semilinear morphisms: f ∈ MorSchK,L (A, X) and g ∈ MorSchK,L (B, Y ). Let C be the product of A and B in SchL, and let q1 ∈ MorSchL (C, A) and q2 ∈ MorSchL (C,B) be the standard projections. In this case we will denote by f × g ∈ MorSchK,L (C,Z) the unique σ-semilinear morphism such that p1|K ◦ (f × g) = f ◦ q1|K and p2|K ◦ (f × g) = f ◦ q2|K . Informally speaking, this is a straightforward way to build a morphism A × B → X × Y out of morphisms A → X and B → Y . After we have this lemma, it is easy to construct a Galois-semiaction on a product of two L-schemes X and Y out of two semiactions on X and on Y . Precisely, if ψ1 : Γ×X|K → X|K and ψ2 : Γ×Y |K → Y |K are two Galois-semiactions, then the new semiaction on Z = X × Y (the product in SchL), which we will call the product of semiactions and denote ψ1 × ψ2, is deﬁned as follows: (ψ1 × ψ2)|{σ}×Z|K =

(ψ1)|{σ}×X|K × (ψ2)|{σ}×Y |K . (In fact, (Z, ψ1 × ψ2) will then be the product of (X, ψ1) and (Y, ψ2) in (StSchL, Γ), but we will not need this.) Using products of semiactions, we can speak about Γ-equivariant morphisms between products of varieties. In particular, if K, L, and Γ are as above, and G is an algebraic group over L, then we call a Galois-semiaction ψ on G compatible with the group structure if the multiplication and inversion map are Γ-equivariant, and the unit is a fixed point of Γ. And if, in addition, (E, ϕ) is a G-torsor with a Galois-semiaction ψ′ on E (and ψ is still a semiaction compatible with the group structure), we call (ψ, ψ′) compatible with the torsor structure if ϕ is Γ-equivariant. Now, using Theorem 2.6 and these notions and still keeping K, L, and Γ as above, one can similarly construct mutually quasiinverse equivalences between the categories of (affine) algebraic groups over a field K and (affine) algebraic groups over L equipped with a compatible Galois-semiaction. By a slight abuse of notation, we also denote these functors by ·L,Γ and DecK . With torsors one should be a bit more careful, because for groups we have only quasiinverse equivalences, and if G is a group over K, then DecK (GL,Γ) is a different group, even though canonically isomorphic. So, for torsors we get equivalences of categories, but not mutually quasiinverse equivalences: their composition is a functor from G-torsors to DecK (GL,Γ)-torsors. However, we still denote these equivalences by ·L,Γ and DecK . This finishes the part of theory of Galois descent that we need. To apply this theory to torsors, we start with a few lemmas. Lemma 2.8. Let (E, ϕ) be a torsor of an algebraic group G. If e is a rational point of E, then the map trive = ϕ|G×{e} : G → E is an isomorphism. Proof. Well-known.

5 We keep the notation trive until the end of the text and call a torsor trivial if it has a rational point. Lemma 2.9. Let (E, ϕ) be a torsor of a smooth algebraic group G over a ﬁeld K. Then there exists a ﬁnite Galois extension L of K such that (EL, ϕL) is a trivial GL-torsor.

′ Idea of the proof. Find a finite extension L such that EL′ has a rational point. Then EL′ is isomorphic to GL′ , therefore smooth, and E is smooth. Smooth schemes obtain a rational point after scalar expansion to a separable closure ([17, Prop. 3.2.20]). Using finite type, we can keep only a finite (automatically still separable) subextension and keep the rational point. So, instead of studying a torsor without rational points, we can make a finite Galois extension of scalars and study a torsor with a rational point and with a compatible Galois-semiaction. From now on, we fix until the end of this section: a split semisimple algebraic group G over a field K, a Borel subgroup B of G, a G-torsor (E, ϕ), a finite Galois extension L of K such that EL has a rational point, and a rational point e ∈ EL. Denote Γ = Gal(L/K). It is known that GL is also split semisimple, and that BL is a Borel subgroup. For any rational point g ∈ GL, denote by Inn g : GL → GL the conjugation by g (so that on rational −1 points, (Inn g)h = ghg ). Denote by ψ1, ψ2, ψ2 the semiactions such that GL,Γ = (GL, ψ1), EL,Γ = (EL, ψ2), and (E/B)L,Γ = ((E/B)L, ψ2). Recall that E/B is defined as a categorical quotient (E × G/B)/G, denote the standard projection E × G/B → E/B by p. It is known (stated in [6, Proposition 12.2]) and can be proved using Galois descent theory that then pL : (EL×(G/B)L) → (E/B)L is also a categorical quotient modulo GL. Denote trivQuote = pL|{e}×(G/B)L . Denote by inv: GL → GL the inversion map. Denote by ϕ: GL ×(E/B)L → −1 (E/B)L the following map: ϕ = pL ◦ (trive ×(trivQuote) ) ◦ (inv × id(E/B)L ).

Lemma 2.10. The map ϕ is an action of GL on (E/B)L. The map trivQuote is a GL-equivariant isomorphism for the (expansion of scalars of) the standard action of G on G/B and for the action ϕ on (E/B)L.

Proof. To see that trivQuote is an isomorphism, use the universal property of categorical quotient. The rest is direct computation. Details omitted.

We keep the notation ϕ and trivQuote until the end of the section. Note that in particular this lemma implies that (E/B)L has a rational point, and this is probably a good point to make a sort of side remark that we will need later about the converse statement. First, let us recall a definition from [9]. Let X be a scheme over an arbitrary field F . The determi- nation function associated with X (see [9, Section 2]) is the following functor from the category of fields containing F to the category consisting of ∅ and a fixed one-element set {0}: A field F1 is mapped to ∅ {0} if and only if XF1 has a rational point, otherwise F1 7→ .

Second, recall that an algebraic group H over K is called special if all torsors of all groups HK1 , where K1 is a ﬁeld extension of K, are trivial. It is known (see, for example, [11, Section 3 and Theorem 2.1]) that B is special. Now, with these two deﬁnitions, we can say that the following lemma becomes a particular case of [9, Lemma 6.5], namely, for the special group P there being equal to B:

′ Lemma 2.11. For any ﬁeld extension K /K, EK′ has a rational point if and only if (E/B)K′ has a rational point.

Now let us return to the map trivQuote and to the action ϕ. The map ϕ (as well as trive and trivQuote) is, however, not Γ-equivariant for the semiactions of Γ we already have. Let us introduce another semiaction of Γ on GL, denoted by ψ3. Namely, for each σ ∈ Γ, set

−1 −1 ψ3|{σ}×GL = (Inn trive (ψ2(σ, e))) ◦ ψ1|{σ}×GL . (2.12)

Lemma 2.13. The semiaction ψ3 is compatible with the group structure. Proof. Computation. Details omitted.

6 We keep the notation ψ3 until the end of this section.

Lemma 2.14. The map ϕ is Γ-equivariant for the semiactions ψ3 × ψ2 and ψ2.

Proof. Computation. Uses Γ-equivariance of ϕL for the semiactions ψ1 × ψ2 and ψ2. Also uses GL- and Γ-equivariance of pL. Details omitted. Now we are going to use the notion of varieties of Borel subgroups of a non-split semisimple algebraic group. It was introduced in [2, 4.9 and 4.10] in higher generality, as variety of subgroups that belong to a certain class, and in [3, 5.24] for parabolic (including Borel) subgroups. It is known (see the previous two references) that if H is a not necessarily split semisimple algebraic group over an arbitrary field, then its variety of Borel subgroups exists. The precise definition involves a certain Galois descent quotient, but instead of following it precisely, it is more convenient to use properties of varieties of Borel subgroups and then prove that all varieties1 with these properties are isomorphic. First, recall that for every semisimple algebraic group over an arbitrary field, there exists a finite separable extension of the field where the group splits ([15, 2.11 and 17.82]) and therefore obtains a Borel subgroup ([15, 21.30]). Now, to be more specific, given a not necessarily split semisimple algebraic group H over K, the variety of Borel subgroups of H is a certain variety Y over K with an action ξ : H × Y → Y . It is known to have the following property.

Let Ks be a separable closure of K. Then there is a bijection between the set of Borel

subgroups of HKs and the set of rational points of XKs , equivariant for the action of (the set (∗)

of rational points of) HKs on the set of its Borel subgroups by conjugation. Lemma 2.15. Let H be a semisimple algebraic group over K. Let Y be an arbitrary variety satisfying ′ (∗). Then there exists a ﬁnite separable extension K of K such that YK′ has a rational point. Moreover, ′ for every such extension K , HK′ has a Borel subgroup (denote it by B0), and YK′ is HK′ -equivariantly isomorphic to HK′ /B0.

Proof. Easy to see. Consider the stabilizer of a rational point of YK′ . The surjectivity of the orbit map follows from the completeness of HK′ /B0 and from the absolute reducedness of Y , see [17, 3.2.20]. Details omitted. Lemma 2.16. Let H be a semisimple algebraic group over K. Let Y be the variety of Borel subgroups of H and let X be a scheme over K with an action of H satisfying the following property:

There exists a ﬁnite separable extension K′ of K such that H ′ is split (denote a Borel K (∗∗) subgroup by B0), and XK′ is HK′ -equivariantly isomorphic to HK′ /B0. Then there exists a unique H-equivariant isomorphism between X and Y .

′ Idea of proof. By Lemma 2.15, w.l.o.g., XK′ has a rational point. Also w.l.o.g., K /K is ﬁnite Galois. ′ Using NHK′ (B0) = B0 and [15, 25.9], check that HK /B0 has no equivariant automorphisms except identity. Then use Theorem 2.6 for K′/K. In other words, this lemma says that up to a unique equivariant isomorphism, (∗∗) can be used as a deﬁnition of the variety of Borel subgroups. Now set H = DecK (GL, ϕ3) and keep this notation until the end of the section. Using Lemma 2.10, we get an action of H on E/B via DecK (ϕ) and the canonical isomorphism E/B =∼ DecK ((E/B)L,Γ). Lemma 2.17. Under this notation, H is a semisimple algebraic group, and E/B with the action of H described above is H-equivariantly isomorphic to the variety of Borel subgroups of H. Proof. The fact that H is semisimple is well-known. The rest follows from Lemmas 2.10 and 2.16.

1We use the word “variety” now as it is used in [2], namely as an absolutely reduced scheme ([2, 2.11]), we don’t mean the elementary version of algebraic geometry where aﬃne varieties are just subsets of vector spaces, without any irrational points.

7 From now on, we suppose that G is simply connected. Then GL and H are known to be simply connected as well. We also assume that L is large enough for all points of Z(GL) (and therefore of Z(HL)) to be rational. Now we will apply the results of [18] and [12] to H and E/B. First, recall that given any two groups G0 and H0 over K (in our case they will be G and H) that become isomorphic over L, there 1 is a standard way to deﬁne an element of H (Γ, Aut((G0)L)) called the cohomology class twisting G0 into H0. More precisely, if (G0)L,Γ = ((G0)L, ζ1) and (H0)L,Γ is isomorphic to ((G0)L, ζ2), then this −1 cohomology class maps each σ ∈ Γ to ζ2|{σ}×(G0)L ◦ (ζ1|{σ}×(G0)L ) . If this class turns out to be in 1 H (Γ, (G0)L/Z((G0)L)), where (G0)L/Z((G0)L) acts on (G0)L by conjugation, then H0 is called an inner form of G0. It follows from formula (2.12) that H is an inner form of G. Denote the cohomology class twisting 1 ∗ Γ G to H by ξ ∈ H (Γ, (G0)L/Z((G0)L)). The paper [18] deﬁnes a map βH,K : ((Z(HL)) ) → Br(K) for a semisimple group H and gives a cohomological interpretation of this map if H is an inner form of a quasi-split (including split) semisimple group G.

∗ Γ Lemma 2.18. The map βH,K : ((Z(HL)) ) → Br(K) deﬁned in [18, 3.5] is zero for the group H we have constructed2. Idea of proof. Following [18, 4.3], consider the exact sequence

′ ′ 1 → T L → ( T L × GL)/Z(GL) → GL/Z(GL) → 1,

′ where T is a split maximal torus of G, and T L is a copy of TL. A direct computation using formula 1 2 ′ (2.12) shows that for the long exact sequence map δ : H (Γ, GL/Z(GL)) → H (Γ, T L), we have δ(ξ) = 0. The rest follows from [18, End of 4.3].

Proposition 2.19. The map of Brauer groups ·⊗K K(E/B): Br(K) → Br(K(E/B)) is injective. Proof. By Lemma 2.17, we can use [12, Theorem B] for the group H. This theorem says that the kernel ker(·⊗K K(E/B)) equals what is denoted in [12] by A(Σ), i.e., the subgroup of Br(K) generated by the images of certain characters on Z(HL) under βH,K . Lemma 2.18 completes the proof. Now we will need a few general lemmas about extension of scalars on schemes (not necessarily on torsors). First, note that for any Galois-semiaction on an irreducible scheme Y there is a straightforward way to extend this semiaction to an action on the set of open subsets of Y , on the field of rational functions on Y , and therefore on the Picard group of Y . Let us recall a well-known result about Picard and Brauer ′ ′ ′ ′ ′ ′ groups. For any two fields K ⊆ L , denote BrL′ (K ) = ker(·⊗K′ L : Br(K ) → Br(L )). Lemma 2.20. Let X be a complete smooth connected scheme over a field K′, and let L′ be a finite Galois extension of K′. Then:

Γ 1. The image of the map ·L′ : Pic(X) → Pic(XL′ ) is contained in Pic(XL′ ) . 2. There is an exact sequence

′ ·L′ Γ ′ ·⊗K (X) ′ 0 → Pic(X) −−→ Pic(XL′ ) → BrL′ (K ) −−−−−−→ Br(K (X))

Proof. Well-known.

Lemma 2.21. Let F1 ⊆ F2 be a finite Galois extension of fields. Let F3 be another extension of F1 such that F4 = F2 ⊗F1 F3 (understood as a tensor product of algebras) is a field. Then F3 ⊆ F4 is a finite Galois extension, Γ = Gal(F4/F3) preserves (normalizes) F2, and the action of Γ on F2 is exactly the action of the Galois group of the extension F1 ⊆ F2.

Moreover, let X be an irreducible and reduced scheme over F1 such that XF2 , XF3 , and XF4 is also ·F4 ∼ ∼ irreducible and reduced. Then the composition Pic(XF2 ) −−→ Pic((XF2 )F4 ) = Pic(XF4 ) = Pic((XF3 )F4 ) is Γ-equivariant. 2In [18], the base ﬁeld is expanded to a separable closure instead of a ﬁnite Galois extension. But since both G and H split over L and all points of Z(HL) are rational, the absolute Galois group of K actually acts via its quotient Γ.

8 Proof. The ﬁrst part is well-known. The second part is direct check. Now we are ready to prove Proposition 2.2 Lemma 2.22. Proposition 2.2 is true if E is a trivial torsor. Proof. Follows from Lemma 2.10 and the explicit description of Pic(G/B) as a free abelian group in Schubert divisors.

Lemma 2.23. Proposition 2.2 is true when K1 equals L (the ﬁeld we ﬁxed earlier in this section).

Proof. One checks easily that (E/B)K(E/B) has a rational point and that L ⊗K K(E/B)= L((E/B)L). By Lemma 2.21, Γ = Gal(L((E/B)L)/K(E/B)).

By Lemma 2.11, EK(E/B) has a rational point. By Lemma 2.22 for the torsor EK(E/B) and the extension L((E/B)L)/K(E/B), we see that

·L((E/B)L) : Pic((E/B)K(E/B)) → Pic((E/B)L((E/B)L))

′ ′ is an isomorphism. By Lemma 2.20 (1) for K = K(E/B) and L = L((E/B)L), the Γ-action on

Pic((E/B)L((E/B)L)) is trivial. Now, also by Lemma 2.22, this time for the torsor EL and the extension L((E/B)L)/L, we get that

·L((E/B)L) : Pic((E/B)L) → Pic((E/B)L((E/B)L)) is an isomorphism. By Lemma 2.21, the Γ-action on Pic((E/B)L) is trivial. Finally, by Proposition ′ ′ 3 2.19 and by Lemma 2.20 (2) for K = K, L = L, we get that ·L : Pic(E/B) → Pic((E/B)L) is an isomorphism. Idea of proof of Proposition 2.2 in the general case. We omit the details regarding commutativity of the diagrams of Picard groups for consecutive field extensions. First, prove the proposition for K1 containing L using Lemma 2.22 for the torsor EL and for the extension K1/L. Then, for a completely arbitrary K1 containing K, we first find a finite Galois extension L1 of K1 for the GK1 -torsor EK1 in the same way as we found and fixed L for K, G, and E. Since L is a finite Galois extension of K, we can construct a field L2 admitting embeddings of L and L1. By the previous ∼ step for EK1 instead of E, Pic((E/B)K1 ) = Pic((E/B)L2 ). By the previous step for the original E, ∼ Pic(E/B) = Pic((E/B)L2 ). Therefore, Pic(E/B) → Pic((E/B)K1 ) is an isomorphism.

3 Estimate of canonical dimension

The next steps of the proof of Theorem 1.4 follows the idea of proof of [10, Proposition 5.1]. First, we will need a well-known fact about the Chow ring of a smooth scheme.

Proposition 3.1. Let X be a smooth scheme over a ﬁeld K, and let L be an extension of K. The map 4 of Chow rings CHL : CH(X) → CH(XL), [Y ] 7→ [YL] for each irreducible and reduced subscheme Y of X is well-deﬁned and is a morphism of rings. Moreover, the isomorphism Pic(X) → CH1(X) commutes with expansion of scalars. Proof. Well-known.

3The idea of using the exact sequence of Brauer and Picard groups to prove the isomorphism between Picard groups is present in [10, Proof of Theorem 1.4]. 4 We will not need this, but the map deﬁned this way actually maps the class of any subscheme Y of X to [YL] ∈ CH(XL).

9 We will also need the following theorem. It is stated in [10, Theorem 2.3] and follows from [7, Corollary 12.2], the preceding commutative diagram, and the definition of distinguished varieties in [7]. More precisely, this definition implies that in the particular case of the commutative diagram, the distinguished varieties are subvarieties of the intersection of supports of the cycles. Recall that a cycle (a formal linear combination of irreducible subvarieties) is called nonnegative if the coefficients in this linear combination are nonnegative, and an element of the Chow ring is called nonnegative if it can be represented by a nonnegative cycle. Theorem 3.2. Let X be a smooth scheme over an arbitrary field K such that the tangent bundle is generated by global sections. Let α and β be nonnegative elements of CH(X). If α (resp. β) is represented by a nonnegative cycle with support on A ⊆ X (resp. B ⊆ X), then αβ can be represented by a nonnegative cycle with support on A ∩ B. We need two more facts from [10]: Lemma 3.3 ([10, Remark 2.4]). Let G be a split simple simply connected algebraic group over an arbitrary field K, let B be a Borel subgroup of G, let E be a G-torsor. Then the tangent bundle of E/B is generated by global sections. Lemma 3.4 ([10, Corollary 2.2]). Let X be a smooth absolutely irreducible scheme over an arbitrary 1 1 field K, and let L be an extension of K. Let α ∈ CH (X). If CHL(α) ∈ CH (XL) is nonnegative, then α ∈ CH1(X) is nonnegative. The following proposition is like Proposition 5.1 in [10], but in a different situation. It is known that if an algebraic group G over a field F is semisimple, split, and simply connected, and B is a Borel subgroup, then for every extension K of F , GK is also semisimple, split, and simply connected, and BK is a Borel subgroup. Proposition 3.5. Let G be a semisimple split simply connected algebraic group over an arbitrary field F . Let B be a Borel subgroup, and let D1,...,Dr ⊂ G/B be the Schubert divisors. Suppose that the n1 nr product [D1] ... [Dr] is multiplicity-free. Let K be a field extension of F , and let E be a GK -torsor. Then there exists a closed, irreducible, and reduced subscheme Y of E/BK of codimension n1 +...+nr such that YK(E/BK ) has a rational point.

Proof. Denote X = E/BK and L = K(X). Write

[D ]n1 [D ]n2 ... [D ]nr = C [Z ]. 1 2 r X w,n1,...,nr w ′ Fix an element v ∈ W such that Cv,n1,...,nr = 1. Set v = vw0. Then it follows from [4, §3.3, Proposition n1 nr n1 nr 1a] that [D1] ... [Dr] [Zv′ ] = [pt]. By Proposition 3.1, we have [(D1)L] ... [(Dr)L] [(Zv′ )L] = [pt] ∈ CH((G/B)L). It is easy to see that XL has a rational point. By Lemma 2.11, EL also has a rational point. Then by Lemma 2.10, XL is isomorphic to (G/B)L. Fix one such isomorphism (it depends on the choice of a rational point of EL) and denote it by f : XL → (G/B)L. Denote the composition f∗ ◦ CHL : CH(X) → CH(XL) → CH((G/B)L) by g. Denote g1 = g|CH1(X). 1 1 By Proposition 2.2 (and by Proposition 3.1), g1 is an isomorphism between CH (X) and CH ((G/B)L) −1 1 For each i (1 ≤ i ≤ r), denote αi = g1 ((D1)L) ∈ CH (X). By Lemma 3.4, these are nonnegative classes (although we don’t claim that each αi is representable by a single irreducible and reduced divisor). n1 nr By Theorem 3.2, the class α1 ...αr is nonnegative. Choose irreducible subvarieties Yi ⊆ X of n1 nr codimension n1+...+nr such that α1 ...αr can be written as their linear combination with nonnegative coeﬃcients. Denote these coeﬃcients by ci ≥ 0:

αn1 ...αnr = c [Y ]. 1 r X i i

It is clear from the deﬁnitions that for each i, g([Yi]) is a linear combination of the irreducible components of f((Yi)L) with nonnegative coeﬃcients. Since g is a morphism of rings (Proposition 3.1), we have n1 n g c [Y ] [(Z ′ ) ] = [(D ) ] ... [(D ) ] r [(Z ′ ) ] = [pt]. X i i v L 1 L r L v L

10 On the other hand, g( ci[Yi])[(Zv′ )L]= (cig([Yi])[(Zv′ )L]), and by Theorem 3.2, each g([Yi])[(Zv′ )L] is (can be written as)P a linear combinationP of (reduced) 0-dimensional subvarieties (i. e. closed points) of f((Yi)L) ∩ (Zv′ )L with nonnegative coefficients. So, a rational point of (G/B)L is equivalent in the Chow ring to a linear combination of some closed points with nonnegative coefficients. Then it follows from the well-definedness of the degree map dim(G/B) CH ((G/B)L) → Z (see [7, Definition 1.4]) that the linear combination actually consists of just one point with coefficient 1, and this point is rational. Recall that this was a point in some intersection f((Yi)L) ∩ (Zv′ )L. In particular, we see that for one of the schemes Yi, (Yi)L has a rational point, and we can set Y = Yi. (We don’t need this, but for this index i we also get ci = 1, and for all other indices i we get g([Yi])[(Zv′ )L]=0or ci = 0.) (Last steps of the) Proof of Theorem 1.4. Let F be the base field of G. It is known that for any extension K of F , GK is also semisimple, split, and simply connected, and BK is a Borel subgroup of GK . We are going to use some results from [9]. As we already mentioned in the Introduction, the definitions of canonical dimension in [9] are not literally the same as here, so for simplicity of notation, we write cd without subscript for the canonical dimension of a scheme in the sense of [9, Section 2] and cd (also without subscript) for the canonical dimension of a group in the sense of [9, Introduction]. As we also mentioned in Introduction, if E is a torsor of a split reductive group, then cd0(E) = cd(E) by [13, Theorem 1.16 and Example 1.18]. Therefore it follows from the statements of Definition 1.3 and the definition of cd in [9, Introduction], that cd(G)= cd0(G) since GK is (in particular) a split reductive group for any extension K of F . Until the end of this paragraph, let K be an extension of F , let E be a GK -torsor. Denote L = K(E/BK). By Proposition 3.5, there exists a subscheme Y ⊆ E/BK of codimension n1 + ... + nr such 5 that YL has a rational point. By [9, Corollary 4.7], we have cd(E/BK ) ≤ dim(G/B) − n1 − ... − nr. Now, [9, discussion after Lemma 6.7] says that cd(G) can be computed as the supremum of cd(E/BK ) for all extensions K of F and all GK -torsors E. Therefore, cd(G) ≤ dim(G/B) − n1 − ... − nr and also cd0(G) ≤ dim(G/B) − n1 − ... − nr.

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